A method for constructing Lyapunov functionals (Q2703879)

The difference system NEWLINE\[NEWLINEx_{i+1}= F(i,x_{-h},\dots,x_i), \quad i\in\mathbb{Z}, \quad F(i,0,\dots,0) =0\tag{1}NEWLINE\]NEWLINE is considered.NEWLINENEWLINENEWLINETheorem. Suppose that there exists a non-negative functional \(V_i=V(i,x_{-h}, \dots, x_i)\), \(i\in \mathbb{Z}\), satisfying the conditions NEWLINE\[NEWLINEV(0,x_{-h},\dots,x_0)\leq c_1 \|\varphi \|^2\quad \text{and} \quad\Delta V_i\leq -c_2|x_i|^2,\;i\in\mathbb{Z},NEWLINE\]NEWLINE where \(\Delta V_i=V_{i+1} -V_i\), \(c_n>0\). Then the zero solution \(x_i\equiv 0\) is asymptotically stable.NEWLINENEWLINENEWLINEConsider the scalar system with a finite delay NEWLINE\[NEWLINEx_{i+1}=ax_i+ b\sum^k_{j=1} (k+1-j)x_{i-j}. \tag{2}NEWLINE\]NEWLINE The inequality NEWLINE\[NEWLINE\bigl(|a|+ |b|k(k+1)/2\bigr) <1NEWLINE\]NEWLINE is a sufficient condition for the asymptotic stability of the zero solution of equation (2).NEWLINENEWLINENEWLINEAn investigation of Volterra equations and stochastic difference systems is carried out.